Let $B$ be a finite dimensional $C^\star$ algebra, and $B\subseteq \mathcal{B(H)}$ for some finite-dimensional Hilbert space $\mathcal{H}$. By $B'$, let us denote the commutant of $B$ in $\mathcal{B(H)}$. I am told that the space of all $B'-B'$ bimodule maps on $\mathcal{B(H)}$ (i.e., from $\mathcal{B(H)}$ to $\mathcal{B(H)}$) can be identified with $B\otimes B^{op}$, where $B^{op}$ is the opposite algebra of $B$, and this is the statement I am trying to prove (unsuccessfully, as yet). Whether this is identification is a linear isomorphism, or an algebra isomorphism, I am yet unsure, but I am leaning towards the latter.
My attempts have mostly been in the following direction: I know that $B$ is faithfully represented on $\mathcal{B(H)}$. This immediately yields an injective homomorphism from $B\otimes B^{op}$ to $\mathcal{B(H)}\otimes \mathcal{B(H)}^{op}$. Now, the space of linear maps from $\mathcal{B(H)}$ to $\mathcal{B(H)}$ is isomorphic (as vector spaces) to $\mathcal{B(H)}\otimes \mathcal{B(H)}^{op}$ by the Choi-Jamiołkowski isomorphism: for all linear maps $A$ from $\mathcal{B(H)}$ to $\mathcal{B(H)}$, $A\mapsto \frac{1}{\dim(\mathcal{H})^2} ( \mathbb{1} \otimes A ) m^\star (\mathbb{1})$, where $m^\star$ is the adjoint of the multiplication operator with respect to the inner product induced by the normalized trace on $\mathcal{B(H)}$, that is $m^\star (\mathbb{1})=N \sum_{i,j=1}^N E_{ij}\otimes E_{ji}$, where $\mathcal{H}\cong \mathbb{C}^N$, and $E_{ij}$'s are the matrix units. But, Choi-Jamiołkowski is not multiplicative (though linear), and it is here that I realize that this approach will not yield an algebra isomorphism.
I am not sure if this holds in the infinite dimensional case, perhaps one has to bring in some complete-boundedness assumption for the $B'-B'$ bimodule maps and / or some restrictions / specifications on $\otimes$ then. But I am not sure still, and in this case, am interested specifically in the finite dimensional case. I am familiar with some (basic) $C^\star$ algebra theory, but not familiar with some advanced topics in rings / modules.